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Online Appendix to the Paper? “Computing Degree of Parallelism for BPMN Processes??”
"... parallelism of P can be computed in O(E  log V  + EL) time where L is the sum of durations of all task nodes in P. In this appendix, we sketch the key ideas in proving Theorem 4.2. Given a process P = (V, s,T, E, τ, δ), an event point is defined as e = (t,max,GE), where t and max are integers, ..."
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parallelism of P can be computed in O(E  log V  + EL) time where L is the sum of durations of all task nodes in P. In this appendix, we sketch the key ideas in proving Theorem 4.2. Given a process P = (V, s,T, E, τ, δ), an event point is defined as e = (t,max,GE), where t and max are integers, GE is a multiset whose elements are pairs of form (g, e′), g ∈ V, g is a join or merge gateway node, and e ′ ∈ E. An event point list is an ordered list whose elements are event points sorted by the first component (t) of each event point in an increasing order. The event point list records the information of a process. Suppose we have an event point list eList = (e1, e2,..., en) where for each ei, ei = (ti,maxi,GEi) is an event point. Since eList is sorted according to ti, it is easy to see that t1 < t2 <... < tn. eList captures two pieces of information: (1) From timestamp ti to ti+1, the cardinality of the corresponding segment’s enactment is maxi; and (2) At timestamp ti, the corresponding segment will invoke exit node g from edge e where (g, e) ∈ GE. Example A.1 In Fig. 4(a) of the original paper, the corresponding event point list is
Quantifying the Parallelism in BPMN Processes using Model Checking
"... A business process is a set of structured, related activities that aims at fulfilling a specific organizational goal for a customer or market. An important metric when developing a business process is its degree of parallelism, i.e., the maximum number of tasks that are executable in parallel in th ..."
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A business process is a set of structured, related activities that aims at fulfilling a specific organizational goal for a customer or market. An important metric when developing a business process is its degree of parallelism, i.e., the maximum number of tasks that are executable in parallel in that process. The degree of parallelism determines the peak demand on tasks, providing a valuable guide for the problem of resource allocation in business processes. In this paper, we investigate how to automatically measure the degree of parallelism for business processes, described using the BPMN standard notation. We first present a formal model for BPMN processes in terms of Labelled Transition Systems, which are obtained through process algebra encodings. We then propose an approach for automatically computing the degree of parallelism by using model checking techniques and dichotomic search. We implemented a tool for automating this check and we applied it successfully to more than one hundred BPMN processes.